Sure, let's analyze the given system of equations step-by-step:
1. We are given the following system of equations:
[tex]\[
\begin{array}{l}
y = -3x + 2 \\
y = 2x - 3
\end{array}
\][/tex]
2. To find the solution, we need to determine the point(s) where both equations intersect. This involves solving the system simultaneously.
3. Since both equations are set equal to [tex]\( y \)[/tex], we can set the right-hand sides of the equations equal to each other:
[tex]\[
-3x + 2 = 2x - 3
\][/tex]
4. Next, we solve for [tex]\( x \)[/tex] by combining like terms and isolating [tex]\( x \)[/tex]:
[tex]\[
-3x + 2 = 2x - 3
\][/tex]
Add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
2 = 5x - 3
\][/tex]
Add [tex]\( 3 \)[/tex] to both sides:
[tex]\[
5 = 5x
\][/tex]
Divide both sides by [tex]\( 5 \)[/tex]:
[tex]\[
x = 1
\][/tex]
5. Now that we have [tex]\( x = 1 \)[/tex], we substitute this value back into one of the original equations to find [tex]\( y \)[/tex]. Using the first equation:
[tex]\[
y = -3(1) + 2 = -3 + 2 = -1
\][/tex]
6. Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (1, -1)
\][/tex]
7. Since we have found a unique solution ([tex]\( x = 1 \)[/tex] and [tex]\( y = -1 \)[/tex]), this indicates that the system has exactly one solution. This means that the equations are consistent (they intersect at exactly one point).
8. Additionally, because the system has a unique solution, it is independent. The lines intersect at one distinct point, rather than being the same line (dependent) or not intersecting at all (inconsistent).
Thus, the system of equations is Consistent and independent.
